Abstract: A new algorithm inspired from particle swarm optimization method is successfully implemented for flexible job-shop work-shop optimization problems. Its efficiency for solving combinatory problems is comparable to the genetic algorithms one taking into account the Makespan criterion.

Hela Boukef graduated from “Institut Supérieur de Gestion de Tunis” in 2003 and obtained the Master of automatic and signal treatment in 2006 at the “Ecole Nationale d’Ingénieur de Tunis”. She is currently preparing the Ph.D. degree in automatic and computer science within the framework of LAGIS-EC-Lille and LARA-ENIT cooperation. Her research is related to optimization methods for discrete events systems, computer science and operational research.

Mohamed Benrejeb has obtained the Diploma of “Ingénieur IDN” (French “Grande Ecole”) in 1973, the Master degree of Automatic Control in 1974, the PhD in Automatic Control of the University of Lille in 1976 and the DSc of the same University in 1980. He is currently a full Professor at the Ecole Nationale d’Ingénieurs de Tunis and an invited Professor at the Ecole Centrale de Lille. His research interests are in the area of analysis and synthesis of complex systems based on classical and non conventional approaches.

Pierre Borne received the Master degree of Physics in 1967, the Masters of Electronics, of Mechanics and of Applied Mathematics in 1968. The same year he obtained the Diploma of “Ingénieur IDN” (French “Grande Ecole”). He obtained the PhD in Automatic Control of the University of Lille in 1970 and the DSc of the same University in 1976. He became Doctor Honoris Causa of the Moscow Institute of Electronics and Mathematics (Russia) in 1999, of the University of Waterloo (Canada) in 2006 and of the Polytechnic University of Bucarest (Romania). He is author or co-author of about 200 Journal articles and book chapters, and of 34 plenary lectures and of more than 250 communications in international conferences. He has been the supervisor of 68 PhD thesis and is author of 20 books. He is Fellow of IEEE and has been President of the IEEE/SMC society in 2000 and 2001. He is presently Professor “de classe exceptionnelle” at the Ecole Centrale de Lille and director of the French pluriformations national group of research in Automatic Control.

The importance of scheduling has increased in recent years due to the growing consumer demand for variety, reduced product life cycles, changing markets with global competition and rapid development of new processes and technologies [Hu and al, 06].

Scheduling problems are part of strong combinatory optimization problems. Many applications, varying from metallurgy, chemistry, agro-food [Tangour, 06], [Tangour, 06] or pharmaceutical industries [Boukef and al, 06], [Boukef and al, 07] can be treated by using heuristics and metaheuristics for their resolution.

But nowadays, a new optimization method is being used and is given satisfying results. This method proposed by Kennedy and Eberhart [Kennedy and Eberhart, 95] is the Particle Swarm Optimization method.

In fact, in 1995, J. Kennedy and R. Eberhart, motivated by bird flocking observation, proposed a new algorithm for representing social behaviour of artificial agents and, then created the Particle Swarm Optimization (PSO). Since 2000, PSO has been growing rapidly [Liao and al, 07] and has been applied successfully to continuous nonlinear functions, neural networks [Van de Bergh and Engelbrecht, 00], etc.

The PSO functioning makes it classified among iterative methods (progressive approach of finding optimal solution) and stochastic ones. Its aim is to improve existing states by moving partially at random and partially according to some defined rules in order to reach the global solution [Clerc, 99].

Most of the research on PSO took into account continuous optimization problems but the studies on discrete ones and particularly on flow-shop [Lian and al, 06], [Lian and al, 06], [Lian and al, 07] and job-shop [Sha and Hsu, 06], [Xia and Wu, 05] scheduling problems are very few.

The principal scope took into account in these types of scheduling problems is the Makespan minimization, even if some authors are interested in total tardiness [Tasgetiren and al, 04].

The problem treated in this paper is dealing with flexible job-shop work-shop scheduling with Makespan optimization objective.

First, flexible job-shop problems are introduced. Next, particle swarm optimization method is presented and a new algorithm inspired from it is proposed for flexible job-shop scheduling problems. Three examples are, then, treated. The two first ones deal with mono-operation flexible job-shop problem and the third with multi-operation flexible job-shop problem. In the last part, a comparison between the obtained results and those issued from genetic algorithm method is proposed, proving, thus, the performance of this new method.

6. Conclusion

The results obtained by applying our algorithm inspired from particle swarm optimization method on flexible job-shop scheduling problems and illustrated with Gantt diagrams, show the effectiveness of this method which leads to a charge balance of operations on selected machines and a minimization of Makespan.

Comparing this method with genetic algorithms one allows us to validate the use of the PSO in the discrete case.